The Invariant Measures Of A Continuous Map And Its Suspended Semi-flow

In this paper, we study the invariant measures of a continuous map and a continuous semi-flow on a compact metric space. The main results are as follows: (1) There exists an one-to-one correspondence between the invariant measures of mutually topologically equivalent semi-flows, and there also exists an one-to-one correspondence between the invariant measures of a continuous map and that of its suspended semi-flow. (2) As an application of (1), in view of the vagueness in the proof of the theorem 2.1: "a continuous flow is uniquely ergodic on 2-dimensional torus if and only if it has at most one periodic orbit" in [2], we give it a proof.